(pp. 241-261)

Lemma (1.1). Let F(x, k) = (f_{0}(x, k), f_{1}(x, k),…,f_{m}(x, k)) be a vector-valued function with each component f_{j}(x, k), j = 0, 1,…,m being regular on K^{n}× Z. Suppose there is a P_{0}, 0 < P_{0}< 1, such that |F(x,k)|^{P0}is sub-regular, with\mathrm{|F(x,k)| = [\Sigma _{j=0}^{m}|f_{j}(x,k)|^{2}]^{1/2}}. Suppose further that for some P > P_{0}, and A > 0,

(1.2) ∫_{Kn}|F(x, k)|^{P}dx ≤ A < ∞ for all k ∈ Z,

then,

(a)\mathrm{f_{j}(x)=\underset{k\rightarrow \infty }{lim} f_{j}(x,k)exists a.e., j = 0, 1,…,m.

(b)\mathrm{\underset{k\rightarrow \infty }{lim} \int _{K^{n}}|F(x,k)-F(x)|^{P}}dx = 0, with F(x) = (f_{0}(x), f_{1}(x),…,f_{m}(x)).

Moreover,

(c)\mathrm{f_{j}^{*}(x) = sup_{k\epsilon Z}|f_{j}(x,k)| \epsilon L^{P}(K^{n}),j = 0,1,\cdots ,m.}

Proof. Let P_{1}= P/P_{0}> 1....